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On the Semi-regular Frames of Translates by Valizadeh, F.; Rahimi, H.; Gol, R. A. Kamyabi; Esmaeelzadeh, F. is a scholarly article available to read on EtoBox.

What is On the Semi-regular Frames of Translates about?

In this note, we fix a real invertible $d\times d$ matrix $\mathcal{A}$ and consider $\mathcal{A}\mathbb{Z}^d$ as an index set. For $f\in L^2(\mathbb{R}^d)$, let $\Phi^{\mathcal{A}}_{f}:=\frac{1}{|\det \mathcal{A}|}\sum_{k\in \mathbb{Z}^d}|\hat{f}(\mathcal{A}^T)^{-1}(\cdot+k)|^2$ be the periodization of $|\hat{f}|^2$. By using $\Phi^{\mathcal{A}}_{f}$, among other things, we characterize when the sequence $\tau_{\mathcal{A}}(f):=\{f(\cdot-\mathcal{A}k)\}_{k\in \mathbb{Z}^d}$ is a Bessel sequence, frame of translates, Riesz basis, or orthonormal basis. And finally, we construct an example, in which $\tau_{\mathcal{A}}(f)$ is a Parseval frame of translates, but not a Riesz sequence.

Author

Valizadeh, F.; Rahimi, H.; Gol, R. A. Kamyabi; Esmaeelzadeh, F.

Published

2019

Language

EN